Electronic Journal of Qualitative Theory of Differential Equations 2004, No. 12, 1-15;http://www.math.u-szeged.hu/ejqtde/
Countably Many Solutions of a Fourth Order
Boundary Value Problem
Nickolai Kosmatov
Department of Mathematics and Statistics
University of Arkansas at Little Rock
Little Rock, AR 72204-1099, USA
[email protected]
Abstract
We apply fixed point theorems to obtain sufficient conditions for existence of infinitely many solutions of a nonlinear fourth order bound-ary value problem
u(4)(t) =a(t)f(u(t)), 0< t <1, u(0) =u(1) =u′
(0) =u′
(1) = 0,
wherea(t) is Lp-integrable andf satisfies certain growth conditions.
Mathematics Subject Classifications: 34B15, 34B16, 34B18.
Key words: Green’s function, fixed point theorems, multiple solutions, fourth order boundary value problem.
1
Introduction
In this paper we are interested in (2,2) conjugate nonlinear boundary-value problem
u(4)(t) =a(t)f(u(t)), 0< t <1, (1)
u(0) =u(1) =u′
(0) =u′
(1) = 0, (2)
The paper is organized in the following fashion. In the introduction we briefly discuss the background of the problem and give an overview of related results. In section 2 we introduce the assumptions on the inhomogeneous term of (1), discuss the properties of the Green’s function of the homogeneous (1), (2), and state the theorems that we use to obtain our main results presented in sections 3 and 4.
Recently Yao [18] applied Krasnosel’ski˘ı’s fixed point theorem [15] to study the eigenvalue problem
u(4)(t) =λf(u(t)), 0< t <1, u(0) =u(1) =u′
(0) =u′
(1) = 0,
The author obtained intervals of eigenvalues for which at least one or two solutions are guaranteed. For other related results we refer the reader to [5, 7, 17, 19].
Fixed point theorems have been applied to various boundary value prob-lems to show the existence of multiple positive solutions. An overview of numerous such results can be found in Guo and Lakshmikantham [4] and Agarwal, O’Reagan and Wong [1].
The study of sufficient conditions for the existence of infinitely many positive solutions was originated by Eloe, Henderson and Kosmatov in [3]. The authors of [3] considered (k, n−k) conjugate type BVP
(−1)n−ku(n)(t) =a(t)f(u(t)), 0< t <1,
u(i)(0) = 0, i= 0, . . . , k−1, u(j)(1) = 0, j = 0, . . . , n−k−1.
Their approach was based on applications of cone-theoretic theorems due to Krasnosel’ski˘ı and Leggett-Williams [16]. For applications of the latter see Davis and Henderson [2] and Henderson and Thompson [6] and the references therein. Later, in [13, 14], the author obtained infinitely many solutions for the second order BVP
−u′′
(t) =a(t)f(u(t)), 0< t <1, αu(0)−βu′
(0) = 0, γu(1) +δu′
(1) = 0,
function a(t). The study of infinitely many solutions was further developed by Kaufmann and Kosmatov [9, 10] to extend the results [13, 14] to the general case of a(t) ∈ Lp[0,1] for p ≥ 1. In [9], a(t) was taken to possess
countably many singularities (or to be an infinite series of singular functions) and, in [10], a(t) was selected in the form of a finite product of singular functions. It is also relevant to our discussion to mention [8, 11, 12] devoted to BVP’s on time scales and three-point BVP’s.
In this note, we extend the results of Yao [18] (with λ = 1). We also generalize and refine the results of [3] (with n = 4, k = 2) by obtaining sharper sufficient conditions for existence of infinitely many solutions of (1), (2).
2
Auxiliaries and fixed point theorems
The Green’s function of
u(4) = 0 satisfying (2) is
G(t, s) = 1 6
(
t2(1−s)2((s−t) + 2(1−t)s), 0≤t≤s≤1,
s2(1−t)2((t−s) + 2(1−s)t), 0≤s≤t≤1. (3)
Definition 2.1 Let B be a Banach space and let K ⊂ B be closed and
nonempty. Then K is said to be a cone if
1. αu+βv ∈ K for all u, v ∈ K and for all α, β ≥0, and 2. u,−u∈ K implies u≡0.
We let B=C[0,1] with the norm kuk= maxt∈[0,1]|u(t)|. In the sequel of
our note we take τ ∈[0,12) and define our cone Kτ ⊂ B by
Kτ ={u(t)∈ B |u(t)≥0 on [0,1], min
t∈[τ,1−τ]u(t)≥cτkuk}, (4)
where cτ = 23τ4. We define an operatorT: B → B by
T u(t) =
Z 1
0
G(t, s)a(s)f(u(s))ds.
Lemma 2.2 The operator T is completely continuous and T: Kτ → Kτ.
Proof: By Arzela-Ascoli theorem,T is completely continuous.
Now we show that it is cone-preserving. To this end, if s∈[t,1], then
With the estimate above we have
min
Fixed points of T are solutions of (1), (2). The existence of a fixed point ofT follows from theorems due to Krasnosel’ski˘ı and Leggett-Williams. Now we state the former.
Theorem 2.3 Let B be a Banach space and let K ⊂ B be a cone in B.
Assume that Ω1, Ω2 are open with 0∈Ω1, Ω1 ⊂Ω2, and let
T: K ∩(Ω2\Ω1)→ K
(i) kT uk ≤ kuk, u∈ K ∩∂Ω1, and kT uk ≥ kuk, u∈ K ∩∂Ω2, or
(ii) kT uk ≥ kuk, u∈ K ∩∂Ω1, and kT uk ≤ kuk, u∈ K ∩∂Ω2.
Then T has a fixed point in K ∩(Ω2\Ω1).
To introduce Leggett-Williams fixed point theorem we need more defini-tions.
Definition 2.4 The map α is said to be a nonnegative continuous concave
functional on a cone K of a (real) Banach space B provided that α: K →
[0,∞) is continuous and
α(tu+ (1−t)v)≥tα(u) + (1−t)α(v)
for all u, v ∈ K and 0≤t≤1.
Definition 2.5 Let 0 < a < b be given and α be a nonnegative continuous concave functional on a cone K. Define convex sets
Br ={u∈ K | kuk< r} and
P(α, a, b) ={u∈ K |a≤α(u), kuk ≤b}.
The following fixed point theorem due to Leggett and Williams enables one to obtain triple fixed points of an operator on a cone.
Theorem 2.6 Let T: Bc →Bc be a completely continuous operator and let
α be a nonnegative continuous concave functional on a cone K such that α(u)≤ kuk for all u∈Bc. Suppose there exist 0< a < b < d≤c such that (C1) {u∈P(α, b, d)|α(u)> b} 6=∅ and α(T u)> b for u∈P(α, b, d), (C2) kT uk< a for kuk ≤a, and
(C3) α(T u)> b for u∈P(α, a, b) with kT uk> d.
Then T has at least three fixed points u1, u2, and u3 such that ku1k < a,
b < α(u2), and ku3k> a with α(u3)> b
Theorem 2.7 (H¨older) Letf ∈Lp[a, b]withp > 1,g ∈Lq[a, b]withq >1, and 1p + 1q = 1. Then f g ∈L1[a, b] and
kf gk1 ≤ kfkpkgkq
Let f ∈L1[a, b] and g ∈L∞[a, b]. Then f g∈L1[a, b] and
kf gk1 ≤ kfk1kgk∞.
The following assumptions on the inhomogeneous term of (2) will stand throughout this paper:
(A1) f is nonnegative and continuous; (A2) lim
Any fixed points of T are now positive.
We will need to employ some estimates on (3) that are given below. One can readily see that
and its minimum on the interval [τ,1−τ] at t=τ,1−τ so that
Using the fact that (7) attains its minimum on the interval [τ′
, τ′′
] ⊂ (0,12) at one of the end-points and defining
l(τ′
It follows from (6) that
max
t∈[0,1]kG(t,·)k1 =
1
384. (10)
One can also easily see from (5) that
max Remark: Other estimates on (3) (used in construction of cones) can be found in [3, 18].
3
Positive solutions and Krasnosel’ski˘ı’s fixed
where
C= max
384
m(1 + 16(τ14−τ13)),1
.
Assume that f satisfies
(H1) f(z)≤M1Ak for all z ∈[0, Ak], k∈N, where M1 ≤192\kakp. (H2) f(z)≥CBk for all z∈[cτkBk, Bk], where cτk =
2 3τk4.
Then the boundary value problem (1), (2) has infinitely many solutions
{uk}∞k=1. Furthermore, Bk ≤ kukk ≤Ak for each k ∈N.
Proof: For a fixed k, define Ω1,k = {u∈ B : kuk < Ak}. The cone Kτk
is given by (4) with τ =τk. Then
u(s)≤Ak=kuk
for all s ∈[0,1]. By (H1),
kT uk = max
t∈[0,1]
Z 1
0
G(t, s)a(s)f(u(s))ds
≤ max
t∈[0,1]
Z 1
0
G(t, s)a(s)ds M1Ak.
Since p >1, takeq= p−p1 >1. Then, by Theorem 2.7,
kT uk ≤ max
t∈[0,1]kG(t,·)kqkakpM1Ak.
From (11) and (H1),
kT uk < 1
192kakpM1Ak
< Ak.
Since kuk=Ak for all u∈ Kτk∩∂Ω1,k, then
kT uk<kuk. (12)
Remark: Note that since 1 + 16(τ14 −τ13) < 1 and kakp ≥ m, we have
Now define Ω2,k = {u ∈ B : kuk < Bk}. Let u ∈ Kτk ∩∂Ω2,k and let
Theorem 3.2 Let {τk}∞k=1 be such thatτ1 < 12 andτk ↓τ∗ >0. Let{Ak}∞k=1
and {Bk}∞k=1 be such that
Ak+1 < ckBk < Bk< CBk < Ak, k ∈N,
where C and cτk are as in Theorem 3.1.
Assume that f satisfies (H2) and
(H3) f(z)≤M2Ak for all z ∈[0, Ak], k∈N, where M2 ≤384\kak∞.
Then the boundary value problem (1), (2) has infinitely many solutions
{uk}∞k=1. Furthermore, Bk ≤ kukk ≤Ak for each k ∈N.
Proof: We now use (10) and repeat the argument above.
Our last result corresponds to the case of p= 1.
Theorem 3.3 Let {τk}∞k=1 be such thatτ1 < 12 andτk ↓τ∗ >0. Let{Ak}∞k=1
and {Bk}∞k=1 be such that
Ak+1 < ckBk < Bk< CBk < Ak, k ∈N,
where C and cτk are as in Theorem 3.1.
Assume that f satisfies (H2) and
(H4) f(z)≤M3Ak for all z ∈[0, Ak], k∈N, where M3 ≤192\kak1.
Then the boundary value problem (1), (2) has infinitely many solutions
{uk}∞k=1. Furthermore, Bk ≤ kukk ≤Ak for each k ∈N.
Proof: For a fixed k, define Ω1,k ={u∈ B : kuk< Ak}. Then
for all s ∈[0,1]. By (H4) and (5),
kT uk = max
t∈[0,1]
Z 1
0
G(t, s)a(s)f(u(s))ds
≤ max
t∈[0,1]
Z 1
0
G(t, s)a(s)ds M3Ak
≤ Z 1
0
max
t∈[0,1]G(t, s)a(s)ds M3Ak
≤ max
s,t∈[0,1]G(t, s)
Z 1
0
a(s)ds M3Ak
= 1
192kak1M3Ak
≤ Ak
and thus we obtain (12), which together with (13) completes the proof.
4
Positive solutions and Legett-Williams fixed
point theorem
In this section we only consider the case of p > 1. The existence theorems corresponding to the cases of p = 1 and p = ∞ are similar to the next theorem and are omitted.
For our cone we now choose
K={u∈ B |u(t)≥0}
and our nonnegative continuous concave functionals on K are defined by
αk(u) = min t∈[τk,1−τk]
u(t)
with αk(u)≤ kuk for eachτk ∈(0,12). For the rest of the note cτk is denoted
by ck.
Theorem 4.1 Let{τk}∞k=1 be such thatτ1 < 12 andτk ↓τ∗ >0. Let{Ak}∞k=1,
{Bk}∞k=1, and {Ck}∞k=1 be such that
Ck+1 < Ak < Bk<
1
ck
where M1 is as in Theorem 3.1. Suppose that f satisfies
Then the boundary value problem (1), (2) has three infinite families of solutions {u1k}∞k=1, {u2k}∞k=1, and {u3k}∞k=1 satisfying ku1kk < Ak, Bk <
We use (H5) and (H7) and repeat the argument leading to (12) to see that
T: BAk → BAk and T: BCk → BCk. Thus, the condition (C2) of Theorem
2.6 is satisfied.
As in Definition 2.5, set
Now, by (H7), using (7)-(9) we have
the assumption (C1) of Theorem 2.6 is satisfied.
If, in addition, u ∈ P(αk, Bk, Ck) with kT uk > c1
Since all hypotheses of Theorem 2.6 are satisfied, the assertion follows.
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